?

Average Error: 0.02% → 0.02%
Time: 4.9s
Precision: binary64
Cost: 13504

?

\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x}}{1 + x}\right) \]
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (* 2.0 (atan (/ (sqrt (- 1.0 (* x x))) (+ 1.0 x)))))
double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}
↓
double code(double x) {
	return 2.0 * atan((sqrt((1.0 - (x * x))) / (1.0 + x)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(sqrt(((1.0d0 - x) / (1.0d0 + x))))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan((sqrt((1.0d0 - (x * x))) / (1.0d0 + x)))
end function

public static double code(double x) {
	return 2.0 * Math.atan(Math.sqrt(((1.0 - x) / (1.0 + x))));
}
↓
public static double code(double x) {
	return 2.0 * Math.atan((Math.sqrt((1.0 - (x * x))) / (1.0 + x)));
}

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))
↓
def code(x):
	return 2.0 * math.atan((math.sqrt((1.0 - (x * x))) / (1.0 + x)))

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end
↓
function code(x)
	return Float64(2.0 * atan(Float64(sqrt(Float64(1.0 - Float64(x * x))) / Float64(1.0 + x))))
end

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end
↓
function tmp = code(x)
	tmp = 2.0 * atan((sqrt((1.0 - (x * x))) / (1.0 + x)));
end

code[x_] := N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(2.0 * N[ArcTan[N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x}}{1 + x}\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.02

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]

  2. Applied egg-rr0.02

    \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\frac{\sqrt{1 - x \cdot x}}{1 + x}\right)} \]

  3. Final simplification0.02

    \[\leadsto 2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x}}{1 + x}\right) \]

Alternatives

Alternative 1
Error0.02%
Cost13376
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
Alternative 2
Error0.47%
Cost7616
\[2 \cdot \tan^{-1} \left(\frac{\left(-1 - \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(x + -1\right)}{1 - x \cdot x}\right) \]
Alternative 3
Error0.46%
Cost7360
\[2 \cdot \tan^{-1} \left(\left(-1 - \left(x \cdot x\right) \cdot -0.5\right) \cdot \frac{1}{-1 - x}\right) \]
Alternative 4
Error0.46%
Cost7360
\[2 \cdot \tan^{-1} \left(\frac{1}{\frac{-1 - x}{-1 - \left(x \cdot x\right) \cdot -0.5}}\right) \]
Alternative 5
Error0.46%
Cost7232
\[2 \cdot \tan^{-1} \left(\frac{1 + \left(x \cdot x\right) \cdot -0.5}{1 + x}\right) \]
Alternative 6
Error0.67%
Cost7104
\[2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right) \]
Alternative 7
Error1.03%
Cost6976
\[2 \cdot \frac{1}{\frac{1}{\tan^{-1} \left(1 - x\right)}} \]
Alternative 8
Error1.02%
Cost6720
\[2 \cdot \tan^{-1} \left(1 - x\right) \]
Alternative 9
Error2.16%
Cost6592
\[2 \cdot \tan^{-1} 1 \]

Error

Reproduce?

herbie shell --seed 2023102 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))